CR compactification for asymptotically locally complex hyperbolic almost Hermitian manifolds

In this article, we consider a complete, non-compact almost Hermitian manifold whose curvature is asymptotic to that of the complex hyperbolic plane. Under natural geometric conditions, we show that such a manifold arises as the interior of a compact almost complex manifold whose boundary is a strictly pseudoconvex CR manifold. Moreover, the geometric structure of the boundary can be recovered by analysing the expansion of the metric near infinity.


Introduction
The complex hyperbolic space is the unique simply connected, complete, Kähler manifold of constant negative holomorphic sectional curvature (we adopt the convention that this constant is −1).It is the complex analogue of the real hyperbolic space, and similarly to its real counterpart, the complex hyperbolic space can be compactified by a sphere at infinity.This sphere at infinity carries a natural geometric structure, which is closely related to the Riemannian geometry of the complex hyperbolic space: their respective groups of automorphisms are in one-to-one correspondence.This structure is that of a strictly pseudoconvex CR manifold, namely, the CR sphere (S, H, J).If S is thought of as the unit sphere of C N , then H = (T S) ∩ (iT S) is the standard contact distribution, and J is given by the multiplication by i in H. Set ρ = e −r with r the distance function to a fixed point.Then ρ is a defining function for the boundary of the above compactification, and as ρ → 0, the complex hyperbolic metric has the asymptotic expansion 1 with θ the standard contact form of S, and γ = dθ| H×H (•, J•) the associated Levi-form.The strict pseudoconvexity of the boundary means that the Levi-form is positive definite on H.The aim of this paper is to construct a similar compactification by a strictly pseudoconvex CR structure for complete, non-compact, almost Hermitian manifolds satisfying some natural geometric conditions.These conditions are the existence of a convex core (called an essential subset ), the convergence of the curvature tensor R to that of the complex hyperbolic space R 0 near infinity, and the fact that the underlying almost complex structure J is asymptotically Kähler at infinity.More precisely, we show the following.
Main Theorem.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of real dimension at least 4, which admits an essential subset.Let r be the distance function to any compact subset.Assume that there exists a > 1 such that R − R 0 g , ∇J g , ∇R g , and ∇ 2 J g = O(e −ar ).
and the restriction J 0 = J| H0 are of class C 1 .Moreover, H 0 is a contact distribution, and J 0 is formally integrable, and (∂M , H 0 , J 0 ) is a strictly pseudoconvex CR manifold.
In addition, the metric g is asymptotically complex hyperbolic: there exists a defining function ρ for the boundary, a C 1 contact form η 0 calibrating H 0 , and a continuous Carnot metric γ, with η 0 and γ 0 = γ| H0×H0 > 0 of class C 1 , such that if a > 3  2 . (1. 2) The contact form and the Carnot metric are related by the relation This result gives a geometric characterisation of complete, non-compact, almost Hermitian manifolds admitting a compactification by a strictly pseudoconvex CR structure.Notice the similarity between equations (1.1) and (1.2).The real analogue of this result, involving a compactification by a conformal boundary for asymptotically locally real hyperbolic manifolds, has been proven by E. Bahuaud, J. M. Lee, T. Marsh and R. Gicquaud [2,3,4,5,12], pursuing the seminal work of M. T. Anderson and R. Schoen [1].
In a previous paper [14], the author proved a similar result in the Kähler case.The improvement here is twofold.First, we are able to remove the Kähler assumption, which was of great importance in the previous proof.Here, the almost complex structure is no more assumed to be parallel, and in fact, needs not even be formally integrable, nor the associated almost symplectic form needs to be closed.In particular, the result applies to perturbations of asymptotically complex hyperbolic Kähler metrics which are only almost Hermitian.Second, the strict pseudoconvexity of the boundary is obtained with an exponential decay of order a > 1, while the earlier version of this result needed a decay of order a > 3  2 .Note that this has a cost: the Carnot metric can be shown to be C 1 only in the direction of the contact distribution.This is the reason why the extended almost complex structure J is only continuous in the transverse direction.Both improvements imply that the set of examples to which the result applies is much increased.
A compactification by a CR structure for some complete, non-compact, Kähler manifolds was already given by J. Bland [10,11], under assumptions that are rather analytic and not totally geometric.To obtain a continuous compactification with no regularity on the CR structure, these assumptions imply the a posteriori estimates R − R 0 g , ∇R g = O(e −4r ) [1] .A strictly pseudoconvex boundary of class C 1 is obtained under assumptions that imply the even stronger estimates R − R 0 g , ∇R g , ∇ 2 R g = O(e −5r ).It was proven by O. Biquard and M. Herzlich [8] that for asymptotically complex hyperbolic Kähler-Einstein metrics in real dimension 4, the curvature tensor has the form R = R 0 + Ce −2r + o g (e −2r ), where C is a non-zero multiple of the Cartan tensor of the CR boundary.It is known that the Cartan tensor vanishes exactly when the CR structure is locally equivalent to that of the sphere (such CR manifolds are called spherical).Many examples are then not covered by J. Bland's results.
The paper is organized as follows.In Section 2, we set up the notations and explain the main idea of the proof of our main Theorem.In Section 3, we compute the expansion of the metric near infinity and prove the existence of the objects η 0 and γ, see Theorem 3.18.Section 4 is dedicated to prove the existence of J 0 , see Theorem 4.5.At this step, η 0 , γ and J 0 are continuous tensor fields.We show in Section 5 that they have higher regularity and that they induce a strictly pseudoconvex CR structure, see Theorems 5.8, 5.10 and 5.14.Finally, we prove our main Theorem in Section 6. [1] At first, one sees that these assumptions imply that R − R 0 g = O(e −3r ) and ∇R g = O(e −4r ).Since on a Kähler manifold it holds that ∇R 0 = 0, applying Kato's inequality to R − R 0 yields the claimed estimate.

Preliminaries
2.1.Notations.Let (M, g) be a Riemannian manifold.Its Levi-Civita connection is denoted by ∇.Our convention on the Riemann curvature tensor is Besse's convention [6], namely ), for vector fields X, Y and Z.By abuse of notation, we still denote by R its four times covariant version: this means that we write R(X, Y, Z, T ) = g(R(X, Y )Z, T ) for vector fields X, Y , Z and T .With this convention, the sectional curvature of a tangent plane P with orthonormal basis {u, v} is sec Essential subsets and normal exponential map.Following [2,3,5,12], an essential subset K ⊂ M is a codimension 0, compact, totally convex submanifold, with smooth boundary ∂K which is oriented by a unit outward vector field ν, and such that sec(M \ K) < 0. In that case, the normal exponential map is a diffeomorphism.The level hypersurface at distance r above K is denoted by ∂K r .For r 0, E induces a diffeomorphism E r : ∂K → ∂K r given by E r (p) = E(r, p); the induced Riemannian metric E * r g on ∂K is denoted by g r .Gauss Lemma states that E * g = dr ⊗ dr + g r .Note that g 0 = g| ∂K .
The gradient of the distance function r on M \ K, called the radial vector field, is denoted by ∂ r .A radial geodesic is a unit speed geodesic ray of the form r → E(r, p) with p ∈ ∂K.Note that the restriction of ∂ r to a radial geodesic is its tangent vector field: therefore, ∂ r satisfies the equation of geodesics ∇ ∂r ∂ r = 0.More generally, a vector field X on M \ K is called radially parallel if ∇ ∂r X = 0.The shape operator S is the field of symmetric endomorphisms on M \ K defined by SX = ∇ X ∂ r .
The normal Jacobi field on M \ K associated to a vector field v on ∂K is defined by Y v = E * v.Such vector fields are orthogonal to and commute with the radial vector field ∂ r .They satisfy the Jacobi field equation ∇ ∂r (∇ ∂r Y v ) = −R(∂ r , Y v )∂ r , and their restriction to any radial geodesic are thus Jacobi fields.Normal Jacobi fields are related to the shape operator S by the first order linear differential equation Almost Hermitian manifolds.An almost Hermitian manifold (M, g, J) is a Riemannian manifold (M, g) together with an almost complex structure J which is compatible with the metric, in the sense that it induces linear isometries in the tangent spaces: one has g(JX, JY ) = g(X, Y ) for all vector fields X and Y .Note that this implies that J is skew-symmetric (in fact, these two properties are equivalent).A tangent plane P ⊂ T M is called J-holomorphic (respectively totally real ) if JP = P (respectively JP ⊥ P ).The constant −1 J-holomorphic sectional curvature tensor R 0 on (M, g, J) is defined by the equality for X, Y and Z vector fields on M .Similarly to the Riemann curvature tensor, we still denote by R 0 its fully covariant version, meaning that R 0 (X, Y, Z, T ) = g(R 0 (X, Y )Z, T ) for all vector fields X, Y , Z and T .Note that R 0 For any pair of orthogonal unit tangent vectors u and v, R 0 (u, v, u, v) = − 1 4 (1 + 3g(Ju, v) 2 ); the minimal value −1 (respectively the maximal value − 1 4 ) is achieved precisely when {u, v} spans a J-holomorphic plane (respectively a totally real plane).In the specific case of the complex hyperbolic space, R 0 coincides with the curvature tensor of the complex hyperbolic metric (see [13,Section IX.7]).CR manifolds.A CR manifold (for Cauchy-Riemann) is a triplet (M, H, J) where H is a tangent distribution of hyperplanes and J is an almost complex structure on H, such that the distribution is a section of H 1,0 whenever X and Y are).In this case, J is said to be formally integrable.A CR manifold is called strictly pseudoconvex if there exists a contact form η calibrating the distribution H (i.e.H = ker η and dη induces a non-degenerate 2-form on H), and if the associated Levi form dη| H×H (•, J•) is positive definite on H.

2.2.
The asymptotic conditions.Throughout the paper, (M, g, J) will denote a complete, non-compact, almost Hermitian manifold of dimension 2n + 2 4, with an essential subset K.We define the following asymptotic geometric conditions.Definition 2.1 ((ALCH) and (AK) conditions).Let (M, g, J) be a complete, non-compact, almost Hermitian manifold.Let r be the distance function to a compact subset.
Lemma 2.3.Assume that (M, g, J) is a complete, non-compact, almost Hermitian manifold, admitting an essential subset K, and satisfying the (ALCH) condition of order a > 0. Let S = ∇∂ r be the shape operator of the level hypersurfaces above K.Then one has In any case, one has S g = O(1), and exp( r 0 S g − 1) = O(1).We also define the following analogous asymptotic conditions of higher order.Definition 2.4 ((ALCH+) and (AK+) conditions).Let (M, g, J) be a complete, non-compact, almost Hermitian manifold.Let r be the distance function to a compact subset.
(1) We say that (M, g, J) satisfies the (ALCH+) condition of order a > 0 if one has the estimates R − R 0 g = O(e −ar ) and ∇R g = O(e −ar ).(2) We say that (M, g, J) satisfies the (AK+) condition of order a > 0 if one has the estimates ∇J g = O(e −ar ) and ∇ 2 J g = O(e −ar ).
Remark 2.5.Under the (AK) condition of order a > 0, one has ∇R 0 g = O(e −ar ).Thus, under the (AK) condition of order a > 0, Kato's inequality shows that the (ALCH+) condition of order In practice, r will be the distance function to the essential subset K.The constants involved in the previous estimates are global.Moreover, in what follows, all estimates of the form f = O(h) will involve a constant that is global.When built out of the choice of a reference frame (which will soon be called an admissible frame, see Definition 3.2), the constant will be independent of [2] For this condition implies that the local geometry at infinity resembles that of the complex hyperbolic space.that choice.By the expressions Y u g = O( u g0 e r ) or Y u = O g ( u g0 e r ), we mean that there exists C > 0 such that for any vector field u on ∂K, one has ∀r 0, ∀p ∈ ∂K, (Y u ) E(r,p) g C u p g0 e r .

2.3.
Outline of the proof.If (M, g, J) is assumed to be Kähler (that is, if ∇J = 0), the author showed in a previous paper [14] the following result.
Theorem ([14, Theorems A, B, C and D]).Let (M, g, J) be a complete, non-compact, Kähler manifold admitting an essential subset K. Assume that there is a constant a > 1 such that the estimates R − R 0 g , ∇R g = O(e −ar ) hold, where r is the distance function to any compact subset.Then on ∂K, there exist a contact form η of class C 1 , and a continuous symmetric positive bilinear form γ, positive definite on the contact distribution H = ker η, such that E * g = dr 2 + e 2r η ⊗ η + e r γ + lower order terms. (2.2) If moreover a > 3 2 , then γ is of class C 1 , and there exists a C 1 formally integrable almost complex structure J H on H, such that γ| H×H = dη(•, J H •). In particular, (∂K, H, J H ) is a strictly pseudoconvex CR manifold.
(3) If in addition, ∇R g = O(e −ar ), then the family of 1-forms (η r ) r 0 converges in C 1 topology, the limit η is of class C 1 , and is contact.The proof uses several estimates, and tedious computations involving many curvature terms.(4) If a > 3  2 , then (η j r ) r 0 locally uniformly converges in C 1 topology, for any j ∈ {1, . . ., 2n}.Hence, γ is of class , then (ϕ r ) r 0 uniformly converges to a tensor ϕ of class C 1 .Its restriction to H = ker η gives the desired formally integrable almost complex structure J H .
The very first step of the proof crucially relies on the fact that J∂ r is parallel in the radial direction, and in fact, the equality ∇J = 0 is used many times.Note that the Kähler assumption is rather rigid: for instance, one has ∇J = 0 if and only if the 2-form g(J•, •) is closed and J is formally integrable.
In this paper, we extend and improve the results of [14].First, the Kähler condition is removed: in fact, neither the closedness of g(J•, •) nor the formal integrability of J need to be met.We instead consider an almost Hermitian manifold (M, g, J) whose almost complex structure J is only parallel at infinity, by imposing the condition ∇ k J g = O(e −ar ), k ∈ {1, 2}.Second, we show that the strict pseudoconvexity of the boundary can be obtained with a > 1 instead of a > 3  2 .This sharper bound comes from deriving sharp geometric estimates in the direction of the contact structure.
In this context of this paper, the vector field J∂ r is not radially parallel, and one cannot even initiate the above strategy as it stands.The main trick is to prove the existence, under our assumptions, of a unit vector field E 0 on M \ K that is radially parallel, and that satisfies E 0 − J∂ r g = O(e −ar ).This latter vector field is unique.One can then consider a reference frame {E 0 , . . ., E 2n } having nice properties, which we call an admissible frame (see Definition 3.2 below), and try to mimic the above proof.The counterpart is that the computations become longer and more involved; one also needs to show numerous extra estimates.

Metric estimates
This section is dedicated to the derivation of the expansion near infinity of the metric g under the (ALCH) and (AK) conditions.We first define the notion of admissible frames, which simplify future computations.We then derive estimates on the asymptotic expansion of normal Jacobi fields, which turns out to be the main ingredients to show our results.
3.1.Admissible frames.We give a construction for some parallel orthonormal frames along radial geodesics in which later computations will be easier.For v a vector field on ∂K, let V be the vector field on M \ K obtained by the parallel transport of v along radial geodesics.Finally, for r 0, define β r (v) = g(J∂ r , V )| ∂Kr .This defines a family of 1-forms (β r ) r 0 on ∂K.Lemma 3.1.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (AK) condition of order a > 0. Then there exists a continuous 1-form β on ∂K such that Proof.Fix v a vector field on ∂K and r 0. Both ∂ r and V are radially parallel, so that one has β r (v) − β 0 (v) = r 0 ∂ r g(J∂ r , V ) = r 0 g((∇ ∂r J)∂ r , V ).By the (AK) assumption, there exists C > 0 such that ∇J g Ce −ar .The Cauchy-Schwarz inequality now implies that r 0 g((∇ ∂r J)∂ r , V ) r 0 ∇J g V g C 1−e −ar a v g0 .Therefore, (β r (v)) r 0 pointwise converges: let β(v) to be its pointwise limit.It defines a pointwise linear form on the tangent spaces of ∂K, satisfying from which is derived equation (3.1).The convergence is thus uniform, and β is continuous.
We shall now show that β is nowhere vanishing.For all r 0, one has β r g0 = 1 pointwise.Indeed, for any v, Cauchy-Schwarz inequality implies that , where ι r : T ∂K → T ∂K r is induced by the parallel transport along radial geodesics.It follows that β g0 = 1 pointwise, and that β is nowhere vanishing.Definition 3.2.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (AK) condition of order a > 0. Let U ⊂ ∂K be an open subset on which the continuous distribution ker β is trivialisable.Let {e 0 , . . ., e 2n } be an orthonormal frame on U such that β(e 0 ) > 0 and β(e j ) = 0 if j ∈ {1, . . ., 2n}.The associated admissible frame {E 0 , . . ., E 2n } on the cone E(R + × U ) is defined as the parallel transport of {e 0 , . . ., e 2n } along the radial geodesics.
If {E 0 , . . ., E 2n } is an admissible frame, then {∂ r , E 0 , . . ., E 2n } is an orthonormal frame on the cone E(R + ×U ) whose elements are parallel in the radial direction even though they need not be differentiable in the directions that are orthogonal to ∂ r .In the following, we will often refer to admissible frames without mentioning the open subset U ⊂ ∂K on which they are defined.Lemma 3.3.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (AK) condition of order a > 0. Let {E 0 , . . ., E 2n } be an admissible frame.Then β(e 0 ) = 1.
Corollary 3.4.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (AK) condition of order a > 0. Let {E 0 , . . ., E 2n } be an admissible frame and δ be the Kronecker symbol.Then Proof.The first point is a consequence of the equality g(J∂ r , E j ) = β r (e j ) and of equation (3.2).For the second point, notice that from which is derived the claimed estimate.
Remark 3.5.One easily shows that the vector field E 0 is the unique unit vector field X on then ∇ ∂r J∂ r = 0, and thus E 0 = J∂ r .In this specific case, admissible frames can be chosen to be smooth, and correspond to the radially parallel orthonormal frames defined in [14].
Proposition 3.6.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (ALCH) and (AK) conditions of order a > 0. Let {E 0 , . . ., E 2n } be an admissible frame.Then Proof.We prove the first point, the other being shown similarly.One readily verifies from the definition of R 0 that R 0 (∂ r , J∂ r , ∂ r , J∂ r ) = −1, and therefore, it holds that and the result follows from the (ALCH) assumption and from the second point of Corollary 3.4.

Associated coframes and normal Jacobi fields estimates.
Recall that for r 0, the diffeomorphism E r : ∂K → ∂K r is defined by E r (p) = E(r, p).Definition 3.7.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold with essential subset K. Assume that it satisfies the (AK) condition of order a > 0. Let {E 0 , . . ., E 2n } be an admissible frame on the cone E(R + × U ).The associated coframe {η 0 r , . . ., η 2n r } r 0 on U is defined by In any admissible frame, the normal Jacobi field Y v associated to the vector field v on ∂K reads Applying twice the differential operator ∇ ∂r to this last equality, one has Recall that radial Jacobi fields are actual Jacobi fields, which means that they satisfy the second order linear differential equation An identification of the components of ∇ ∂r (∇ ∂r Y v ) in the given admissible frame shows that the coefficients {η j r (v)} j∈{0,...,2n} satisfy the differential system where the functions {u j k } j,k∈{0,...,2n} are defined by Proposition 3.8.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (ALCH) and (AK) conditions of order a > 1 2 .Let {η 0 r , . . ., η 2n r } r 0 be the coframes associated to an admissible frame on U ⊂ ∂K.Then there exists continuous 1-forms {η 0 , . . ., if a > 3 2 .If furthermore one assumes that a > 1, the family {η 0 , . . ., η 2n } is a continuous coframe on U .Corollary 3.9.If a > 1  2 , then η j r g0 is bounded independently of r, j, the choice of an admissible frame, and U .
Proof.For j ∈ {0, . . ., 2n} and r 0, write η j r = η j 0 + r 0 ∂ r η j r .Notice that η j 0 g0 = 1.Then by Proposition 3.8, η j r g0 The following corollary is an immediate consequence of Proposition 3.8.Corollary 3.10.In any admissible frame, the normal Jacobi field Y v associated to a vector field v on ∂K satisfies and As a consequence, one has the global estimates ).
Remark 3.11.Note that although the estimates of Proposition 3.8 are not uniform in all directions, they contribute equally to the lower order term in equations (3.5) and (3.6) thanks to the remaining exponential factors.

3.3.
Global consequences and metric estimates.We shall now highlight global consequences of the study conducted in Subsections 3.1 and 3.2.We then prove the first of our main results.
Lemma 3.12.Assume that (M, g, J) satisfies the (AK) condition of order a > 0. Then the local vector field e 0 defined in Definition 3.2 defines a global continuous vector field on ∂K, independently of the construction of any admissible frame.
Proof.The 1-form β defined in Lemma 3.1 is continuous and nowhere vanishing.Hence, the distribution ker β ⊂ T ∂K is a continuous distribution of hyperplanes.It follows that its g 0orthogonal complement L is a well-defined and continuous line bundle.Notice that the restriction of β trivialises L. It follows that e 0 is the unique section of L that is positive for β, and of unit g 0 -norm.This concludes the proof.
The family of 1-forms {η 0 r } r 0 is then globally defined on ∂K, independently of the choice of the admissible frame.As a consequence, one has the following global version of Proposition 3.8.Proposition 3.13.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, admitting an essential subset K. Assume that it satisfies the (ALCH) and (AK) condition of order a > 1 2 .Then there exists a continuous 1-form η 0 on ∂K such that 2 .If furthermore one assumes that a > 1, then η 0 is nowhere vanishing.
The following Corollary is a straightforward application of the triangle inequality and of Corollary 3.9.
Corollary 3.14.One has the following estimates From Gauss's Lemma, the Riemannian metric g reads as E * g = dr ⊗ dr + g r , with (g r ) r 0 the smooth family of Riemannian metrics on ∂K defined by g r = E * r g.By construction, the first term that appears in the asymptotic expansion of the metric g near infinity is e 2r η 0 ⊗ η 0 .Definition 3.15.For r 0, γ r is defined as γ r = e −r (g r − e 2r η 0 r ⊗ η 0 r ).By definition, (γ r ) r 0 is a family of symmetric 2-tensors on ∂K.Let {η 0 r , . . ., η 2n r } r 0 be the coframes associated to an admissible frame {E 0 , . . ., E 2n }.Then locally, γ r = 2n j=1 η j r ⊗ η j r .Consequently, γ r is positive semi-definite, and is positive definite on ker η 0 r , for any r 0. The following proposition shows that (γ r ) r 0 converges to some tensor that shares similar properties.Proposition 3.16.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, and admitting an essential subset K. Assume that it satisfies the (ALCH) and (AK) conditions of order a > 1 2 .Then there exists a continuous positive semi-definite symmetric 2-tensor γ on ∂K, which we call the Carnot metric, such that If furthermore one assumes that a > 1, then γ is positive definite on ker η 0 .
The previous study implies the following comparison between quadratic forms.Corollary 3.17.If a > 1, there exists a constant λ > 1 such that for all r 0, the comparison between quadratic forms 1 λ e r g 0 g r λe 2r g 0 holds.Proof.For r 0, η 0 r ⊗ η 0 r and γ r are positive symmetric 2-tensors.Define q r = η 0 r ⊗ η 0 r + γ r , which is a Riemannian metric on ∂K.From g r = e 2r η 0 r ⊗ η 0 r + e r γ r , one readily checks that ∀r 0, e r q r g r e 2r q r . (3.8) According to Propositions 3.13 and 3.16, q r uniformly converges to the continuous Riemannian metric q ∞ = η 0 ⊗ η 0 + γ as r → ∞.Let S g0 ∂K be the unit sphere bundle of (∂K, g 0 ), which is compact by compactness of ∂K.The map (r, v) ∈ [0, ∞] × S g0 ∂K → q r (v, v) ∈ (0, ∞) is then continuous on the compact space [0, ∞] × S g0 ∂K.Therefore, there exists λ > 1 such that for all r 0, 1 λ q r λ on S g0 ∂K.The result now follows from equation (3.8) and from the homogeneity of quadratic forms.
We shall now show the first of our main results.Theorem 3.18.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (ALCH) and (AK) assumptions of order a > 1  2 .Then on ∂K, there exists a continuous 1-form η 0 and a continuous positive semi-definite symmetric 2-tensor γ, such that in the normal exponential map E, the Riemannian metric g reads if a > 3  2 . (3.9) If furthermore one assumes that a > 1, then η 0 is nowhere vanishing, and γ is positive definite on the distribution of hyperplanes ker η 0 .
3.4.Estimates on the shape operator.Before we conclude this section, we give another consequence of the previous study: we derive asymptotic estimates on the shape operator S. First, we introduce a natural vector field ξ 0 , which is closely related to S. Definition 3.20.The vector fields (ξ r 0 ) r 0 on ∂K are defined as ξ r 0 = E * r (e r E 0 ).Proposition 3.21.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, admitting an essential subset K. Assume that it satisfies the (ALCH) and (AK) conditions of order a > 1.Then there exists a continuous vector field ξ 0 on ∂K such that (3.10) It is uniquely characterised by the fact that η 0 (ξ 0 ) = 1 and γ(ξ 0 , ξ 0 ) = 0.
Estimates (3.10) now follow from the estimates of Proposition 3.8, together with the fact that ξ 0 g0 is uniformly bounded by continuity of ξ 0 and compactness of ∂K.
For v a vector field on ∂K, the associated normal Jacobi fields Y v satisfies ∇ ∂r Y v = SY v .It follows from equation (3.4) that in an admissible frame, one has For r 0, consider the pull-back S r = E * r S of the shape operator S through the diffeomorphism E r : ∂K → ∂K r .It is well defined since S leaves stable the tangent bundle of the level hypersurfaces ∂K r .Proposition 3.23.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, admitting an essential subset K. Assume that it satisfies the (ALCH) and (AK) conditions of order a > 1 2 .Then the family (S r ) r 0 satisfies the estimates In particular, if a > 1, then S r −→ r→∞ 1 2 (Id +η 0 ⊗ ξ 0 ), and one can substitute η 0 r ⊗ ξ r 0 with η 0 ⊗ ξ 0 in estimates (3.12).
Proof.Let v be a vector field on ∂K.It follows from Proposition 3.8 and from Corollary 3.10 that By the very definition of S r , ξ r 0 and g r , it follows that 2 , Finally, Corollary 3.17 implies that and estimates (3.12) now follow.If a > 1, then estimates on η 0 − η 0 r g0 (Proposition 3.13) and on ξ 0 − ξ r 0 g0 (Proposition 3.21), together with the triangle inequality, show that one can replace η 0 r ⊗ ξ r 0 with η 0 ⊗ ξ 0 in estimates (3.12).This concludes the proof.Remark 3.24.In the complex hyperbolic space, the shape operator of a geodesic sphere of radius r, with outward unit normal ν, is given by S = cotanh(r) Id RJν + 1 2 cotanh( r 2 ) Id {ν,Jν} ⊥ .Proposition 3.23 implies that the local extrinsic geometry of the level hypersurfaces ∂K r is then asymptotic to that of horospheres in the complex hyperbolic space.

The almost complex structure
This section is dedicated to prove the existence of a natural almost complex structure J 0 on the distribution of hyperplanes H 0 = ker η 0 , obtained as the restriction of a naturally defined tensor ϕ on ∂K.
The ambient almost complex structure J does not leave stable the ambient distribution of hyperplanes {∂ r } ⊥ .Consider the orthogonal projection π : T M \ K → T M \ K onto {∂ r } ⊥ .Define Φ to be the field of endomorphisms on M \ K defined by Φ = πJπ.Since π and J have unit norms, then Φ g 1. Formally, one has π = Id −g(∂ r , •) ⊗ ∂ r , and Φ then reads Lemma 4.1.Assume that (M, g, J) satisfies the (AK) condition of order a > 0. For any admissible frame {E 0 , . . ., E 2n } and any vector fields X and Y , one has: (1) Proof.The first point is a straightforward computation.To prove the second point, note that Φ(J∂ r ) = 0, so that Φ(E 0 ) g = Φ(E 0 −J∂ r ) g E 0 −J∂ r g .The result follows from Corollary 3.4.Finally, by the very definition of Φ, Φ(E j ) = JE j − g(E j , J∂ r ), and the last point follows from Corollary 3.4.
The tensor Φ leaves stable the tangent distribution {∂ r , J∂ r } ⊥ .Therefore, one can pull it back through the family of diffeomorphisms (E r ) r 0 .Definition 4.2.The family of endomorphisms (ϕ r ) r 0 is defined by ϕ r = E * r Φ for r 0. Recall that (S r ) r 0 is the family of endomorphisms E * r S induced by the shape operator.Lemma 4.3.Assume that (M, g, J) satisfies the (ALCH) and (AK) assumption of order a > 1.
Then the following estimates hold: if a > 3 2 .Proof.We first show the first point.From Corollary 3.17, there exists c > 0 such that for r 0, ϕ r ξ r . The result now follows from Lemma 4.1 Let us now focus on the second point.Let v be a vector field on ∂K.Corollary 3.17 states that there exists c > 0 such that ϕ r v g0 c Φ(Y v ) g e − r 2 , for all r 0. The result follows from the fourth point of Lemma 4.1.
We are now able to prove that the family (ϕ r ) r 0 converges to a continuous field of endomorphisms, provided that a > 1.
Proposition 4.4.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (ALCH) and (AK) conditions of order a > 1.Then there exists a continuous field of endomorphisms ϕ on ∂K such that if a > 3 2 . (4.1) In addition, ϕ satisfies: (1) Proof.Let us first show the existence of ϕ.The proof goes in two steps.We first derive a differential equation for (ϕ r ) r 0 .Let X be a vector field on M \ K.
Note that the eigenspaces of the projector π are ker π = R∂ r and ker(π − Id) = {∂ r } ⊥ , which are both left stable by the shape operator S. Hence, S commutes with π, from which is derived that that L ∂r Φ = ΦS − SΦ + π(∇ ∂r J)π.Define ψ r = E * r (π(∇ ∂r J)π), so that one has ∂ r ϕ r = ϕ r S r − S r ϕ r + ψ r .A direct application of the (AK) assumption and Corollary 3.17 yields ψ r = O g0 (e −(a− 1 2 )r ).Therefore, it follows from Lemma 4.3 that if a > 3 2 .Consequently, (ϕ r ) r 0 uniformly converges to some continuous tensor ϕ, which satisfies the inequality ϕ r − ϕ g0 = ∞ r ∂ r ϕ r g0 ∞ r ∂ r ϕ r g0 for all r 0. This implies estimates (4.1).Let us now establish the claimed properties satisfied by ϕ.The first two points are immediate consequences of Lemma 4.3.We thus focus on the last claim.One easily checks that Φ satisfies the equality Φ As usual, Corollary 3.17 yields that ǫ r g0 = O(e , where the last equality is due to Corollary 3.4.The first part of the result now follows from the convergence of (η 0 r ) r 0 and of (ξ r 0 ) r 0 when a > 1.The second part of the claim is a consequence of the first point.Proposition 4.4 implies that when a > 1, (∂K, η 0 , ϕ, ξ 0 ) is an almost contact manifold (see [9] for an introduction to this notion).In particular, ϕ induces an almost complex structure on the distribution of hyperplanes H 0 = ker η 0 .The study conducted in this section finally implies the second of our main Theorems.Theorem 4.5.Let (M, g, J) be a complete, non-compact almost Hermitian manifold of dimension greater than or equal to 4 Assume that M satisfies the (ALCH) and (AK) conditions of order a > 1.Let η 0 and γ be given by Theorem 3.18, and let ϕ be defined as in Proposition 4.4.The restriction J 0 = ϕ| H0 of ϕ to the hyperplane distribution H 0 = ker η 0 then induces an almost complex structure, and γ 0 = γ| H0×H0 is J 0 -invariant.

Higher regularity
This section is dedicated to show that under the stronger conditions (ALCH+) and (AK+) of order a > 1, the tensors η 0 , γ, and ϕ defined previously gain in regularity.As a consequence, we highlight a strictly pseudoconvex CR structure related to the expansion of the metric near infinity.
5.1.Order one estimates.We first provide several estimates that will be useful in the following study.
Lemma 5.1.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, admitting an essential subset K. Assume that it satisfies the (ALCH) condition of order a > 1 2 .Let u and v be vector fields on ∂K.Let V be the parallel transport of v along radial geodesics.Then ∇ Yu V = O g ( u g0 v g0 e r ).
2) and that V g = v g0 .Under the (ALCH) condition of order a > 1  2 , one has Y u g = O( u g0 e r ) (Corollary 3.10).The result follows from a straightforward integration.Lemma 5.2.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, admitting an essential subset K. Assume that it satisfies the (ALCH) and (AK) conditions of order a > 1 2 .Then ∇ Yu J∂ r = O g ( u g0 e r ).Proof.Write ∇ Yu J∂ r = (∇ Yu J)∂ r + JSY u .Then ∇ Yu J∂ r g ( ∇J g + S g ) Y u g , and the result follows from Lemma 2.3, the (AK) assumption and the estimates of Corollary 3.10.Lemma 5.3.Assume that (M, g, J) satisfies the (ALCH) and (AK+) conditions of order a > 1 2 .Then ∇ Yu (∇ ∂r J∂ r ) = O g ( u g0 e −(a−1)r ).

Direct computations using the equalities
where R contains all the curvature terms.From this is deduced the almost everywhere inequality ∂ r (e −r π((∇ Yu S)Y v )) g ) e −r R g +( S g −1)(e −r π((∇ Yu S)Y v )) g ).After a straightforward integration, Grönwall's Lemma yields By tensoriality and compactness of ∂K, one has (∇ g u S)v g = O( u g0 v g0 ).Moreover, Lemma 2.3 yields the estimate exp r 0 ( S g − 1) ds = O(1).To conclude, it suffices to show that R = O g ( u g0 v g0 e 3 2 r ).The (ALCH+) assumption of order a > 1  2 yields A close look at the definition of R 0 (see equation (2.1)) shows that the leading terms in R g are of the form cη 0 (u)η j (v)e Let us now show the second point.Similarly, Kato's inequality yields the almost everywhere inequality Straightforward computations, using that ∇ ∂r π = 0, that π and S commute, and that The rest of the proof goes similarly to that of the first point, using the estimates derived on π((∇ Yu S)Y v ) g .The main difference is that the initial data here is not tensorial in v, but instead is π(∇ u v) g = ∇ g0 u v g0 ∇ g0 v g0 u g0 .
Remark 5.5.If one considers the whole vector field ∇ Yu Y v instead, then one only has the estimates ∇ Yu Y v g = O(( v g0 + ∇ g v g ) u g0 e 2r ).Indeed, the radial component is given by when η 0 (u) and η 0 (v) do not vanish.

Regularity of the admissible frames.
We shall now show that under the (ALCH) and (AK+) conditions of order a > 1, the vector field e 0 , defined in Definition 3.2, is actually of class C 1 .
Proposition 5.6.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, admitting an essential subset K. Assume that it satisfies the (ALCH) and (AK+) conditions of order a > 1.Then the vector field e 0 is of class C 1 ; admissible frames can be chosen to have the same regularity.
Proof.It suffices to show that the 1-form β defined in Section 3.1 is of class C 1 .To do so, we shall show that β(v) is a C 1 function for any C 1 vector field v.We prove this later fact by showing that (u(β r (v))) r 0 uniformly converges for any C 1 vector fields u and v on ∂K.Let u and v be such vector fields, and r 0. Then , where V is the parallel transport of v along radial geodesics.Since [∂ r , Y u ] = 0 and ∇ ∂r V = 0, one has , and Y u g = O( u g0 e r ) (Corollary 3.10).It now follows from Lemma 5.1, Lemma 5.3, and the (AK) assumption, that Consequently, ∂ r (u(β r (v))) uniformly converges for any vector fields u and v.This concludes the proof.
It what follows, we will need to differentiate expressions involving ∇ Yu E j in the radial direction, with Y u a normal Jacobi field and E j an element of an admissible frame.At a first glance, this is a priori justified only if E j is of class C 2 .One could prove such regularity by requiring the stronger condition ∇ 3 J g = O(e −ar ).It turns out that one needs not assume this last condition, as a consequence of the fact that E j is solution to the first order linear differential equation ∇ ∂r E j = 0. Indeed, let {r, x 1 , . . ., x 2n+1 } be Fermi coordinates [3] , and write the components of the shape operator S. As a consequence, one can consider elements of the form ∇ ∂r (∇ Yu E j ) even though E j is only of class C 1 .In fact, one has Corollary 5.7.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, admitting an essential subset K. Assume that it satisfies the (ALCH) and (AK+) conditions of order a > 1.Let u be a vector field on ∂K.Then Proof.Let u be a vector field on ∂K, and {E 0 , . . ., E 2n } be an admissible frame of class During the proof of Proposition 5.6, we have shown that (β r ) r 0 converges in C 1 topology.Hence, ∀j ∈ {0, . . ., 2n}, lim 5.3.The contact form and the Carnot metric.We shall now show that if the (ALCH+) and (AK+) conditions of order a > 1 are satisfied, then η 0 and γ| H0×H0 are of class C 1 and that dη 0 (•, ϕ•) = γ.In particular, η 0 is contact.These results are analogous to [14, Theorems B & C], although we give slightly different and considerably shorter proofs here.The main difference is that we prove the C 1 convergence of elements of the form (η j r (v)) r 0 , instead of C 0 convergence of elements of the form (L u η j r ) r 0 .Theorem 5.8.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (ALCH+) and (AK+) conditions of order a > 1.Then η 0 is a contact form of class C 1 .Moreover, dη 0 (•, ϕ•) = γ, and the Reeb vector field of η 0 is ξ 0 .
Proof.The proof is divided in three parts.First, we show that η 0 is of class C 1 .Then we derive an expression for dη 0 (•, ϕ•), and deduce that η 0 is contact.Finally, we show that ξ 0 is the Reeb vector field of η 0 .
To show that η 0 is of class C 1 , we show that for any vector field v, the function η 0 (v) is of class C 1 .To do so, we show that for any other vector field u, (u(η 0 r (v))) r 0 uniformly converges on ∂K.Let u and v be vector fields on ∂K.Let f be the function on M \ K defined by the expression f = e r u(η 0 Similarly, one has One readily checks from the definition of Note that the radial part of ∇ Yu Y v plays no role here due to the symmetries of the Riemann curvature tensor, so that one can substitute ∇ Yu Y v with π(∇ Yu Y v ) in this latter expression.
The next result shows that under the assumptions of Theorem 5.8, the Carnot metric γ 0 on H 0 is of the same regularity.The proof is very similar.Theorem 5.10.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that it satisfies the (ALCH+) and (AK+) conditions of order a > 1.Then γ 0 = γ| H0×H0 is of class C 1 .
Proof.Let {E 0 , . . ., E 2n } be an admissible frame of class C 1 defined on a cone E(R + × U ), and fix j ∈ {1, . . ., 2n}.Let us first show that η j is of class C 1 on the distribution H 0 | U .To do so, we shall prove that u η j r (v) r 0 locally uniformly converges on U for v tangent to H 0 | U and u any vector field on U .
Let u and v be such vector fields, and r 0 be fixed.Let , which is smooth in the radial direction.Since [∂ r , Y u ] = 0 and ∇ ∂r E j = 0, one has and, for Y v is a Jacobi field, one has ).Therefore, one has the equality As in the proof of Theorem 4.5, the radial component of ∇ Yu Y v plays no role due to the symmetries of R, so that one can substitute this term with π(∇ Yu Y v ).Moreover, g(E j , J∂ r ) = β r (e j ), where (β r ) r 0 is the family defined in Section 3.1.Recall that one has the following estimates: • R, S = O g (1) (Remark 2.2 and Lemma 2.3), • R − R 0 , ∇(R − R 0 ) = O g (e −ar ), ((ALCH+) condition and Remark 2.5), • β r (e j ) = O(e −ar ) (Corollary 3.4), Notice that e − r 2 h j = ∂ r (e − r 2 f j ) = ∂ r u(η j r (v)) , from which is deduced that In any case, u(η j r (v)) r 0 locally uniformly converges.As a consequence, η j | H0|U is of class C 1 .We immediately deduce from the local expression γ = 2n j=1 η j ⊗ η j that γ 0 = γ| H0×H0 is of class C 1 .This concludes the proof.
Remark 5.11.With the stronger assumption a > 3  2 , the same proof shows that for j ∈ {1, . . ., 2n}, η j is of class C 1 in all directions, and so is γ.Indeed, in this case, on has to consider the estimate Y v = O g ( v g0 e r ) instead.5.4.The almost complex structure.We shall now show that the almost complex structure J 0 defined on the C 1 distribution H 0 is of the same regularity, and that it is formally integrable.We first remark that the local vector fields {ξ 1 , . . ., ξ 2n } are of class C 1 , although the Reeb vector field ξ 0 might only be continuous.Lemma 5.12.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, with essential subset K. Assume that (M, g, J) satisfies the (ALCH+) and (AK+) conditions of order a > 1.Let {η 0 , . . ., η 2n } be the local coframe associated to any admissible frame {E 0 , . . ., E 2n }.Let {ξ 0 , ξ 1 , . . ., ξ 2n } be its dual frame.Then for j ∈ {1, . . ., 2n}, ξ j is a vector field of class C 1 .
We now show that under the (AK+) condition of order a > 0, admissible frames can almost be chosen to be J-frames, in the following sense.Lemma 5.13.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at least 4, and with essential subset K. Assume that it satisfies the (AK+) condition of order a > 0. Then there exists an admissible frame {E 0 , . . ., E 2n } such that ∀j ∈ {1, . . ., n}, JE 2j−1 − E 2j = O g (e −ar ).
Proof.Let U ⊂ ∂K be an open domain on which H 0 is trivialisable.Let e 1 be a unit section of H 0 | U of class C 1 , and let E 1 be its parallel transport along radial geodesics.Consider the family of 1-forms β 1 r : H 0 | U → R defined by β 1 r (v) = g(V, JE 1 )| ∂Kr , where V is the parallel transport of v along radial geodesics.The same study than that conducted for the proofs of Lemma 3.1 and Proposition 5.6 shows that under the (AK+) condition of order a > 1, there exists a nowhere vanishing 1-form β 1 on U , which is of class C 1 , such that β 1 r − β 1 g0 = O(e −ar ).Let e 2 be the unique C 1 section of H 0 | U such that e 2 ⊥ g0 ker β 1 , e 2 g0 = 1 and β 1 (e 2 ) > 0. Define E 2 to be its parallel transport along radial geodesics.Similarly to Corollary 3.4, one shows that E 2 − JE 1 = O g (e −ar ).The rest of the proof follows by induction.
We refer to such an admissible frame as a J-admissible frame.We are now able to show the last Theorem of this section, exhibiting a strictly pseudoconvex CR structure at infinity.Theorem 5.14.Let (M, g, J) be a complete, non-compact, almost Hermitian manifold of dimension at last 4, with essential subset K. Assume that it satisfies the (ALCH+) and (AK+) condition of order a > 1.Let J 0 be the almost complex structure on H 0 induced by ϕ.Then J 0 is of class C 1 , and is formally integrable.In particular, (∂K, H 0 , J 0 ) is a strictly pseudoconvex CR manifold of class C 1 .
Let us now show that J 0 is formally integrable.Recall that γ| H0×H0 is J 0 -invariant, so that by [14,Proposition 5.10], it suffices to show that N ϕ | H0×H0 = dη 0 | H0×H0 ⊗ ξ 0 , where N A stands for the Nijenhuis tensor of the field of endomorphisms A, defined by where π is the orthogonal projection onto {∂ r } ⊥ .From now, and until the rest of the proof, we assume that u and v are tangent to H 0 .Let r 0, and note that N ϕr = E * r N Φ .The (AK) condition, the uniform bound on S g (Lemma 2.3), estimates on E 0 − J∂ r (Corollary 3.4), estimates on Y u and Y v (Corollary 3.10), comparison between g 0 and g r (Corollary 3.17), and estimates on ϕ r S r − S r ϕ r (Lemma 4.3), now yield the existence of α 1 > 0, depending on a, such that N ϕr (u, v) = e −r (g(Y v , SΦY u ) − g(Y u , SΦY v ))ξ r 0 + O g0 ( u g0 v g0 e −α1r ).Similar calculations that the ones conducted to derive an expression for dη 0 r (u, ϕ r v) (see the proof of Theorem 5.8) show that there exists α 2 > 0 depending on a with e −r (g(Y v , SΦY u ) − g(Y u , SΦY v )) = dη 0 (u, v) + O( u g0 v g0 e −α2r ).
Remark 5.15.If M has dimension 4, then J 0 is an almost complex structure of class C 1 defined on a 2-dimensional vector bundle.Its integrability is automatic in this specific case.
Remark 5.16.Similarly to Remark 5.11, under the stronger assumption a > 3  2 , one shows that ϕ is of class C 1 in all directions.

The compactification
We conclude this paper by showing our main Theorem.
Proof of the main Theorem.We first give a construction for M .Fix K an essential subset and E its normal exponential map.Let M (∞) be the visual boundary of (M, g), which is the set of equivalent classes [σ] of untrapped unit speed geodesic rays σ, where two rays σ 1 and σ 2 are equivalent if and only if the function t 0 → d g (σ 1 (t), σ 2 (t)) is bounded.By is thus a bijection.We endow M with the structure of a compact manifold with boundary through this latter bijection.This identifies M with the interior of M .Note that if ρ > 0, then r = − ln ρ is the distance to K for g in M .A compactly supported modification of ρ in a neighbourhood of K in M provides a smooth defining function for the boundary ∂M = M (∞).By abuse of notation, we still denote it ρ.Let η 0 be the contact form and γ be the Carnot metric given by Theorem 5.8.Let H 0 be the associated contact distribution, and let J 0 be the integrable almost complex structure on H 0 given by Theorem 5.14.We see these objects as defined on ∂M through the diffeomorphism E(0, •) : {0} × ∂K → ∂M .Then (∂M , H 0 , J 0 ) is a strictly pseudoconvex CR manifold of class C 1 by Theorem 5.14.Theorem 3.18 and Remark 3.19 show that the metric g has the desired asymptotic expansion (1.2) near the boundary ∂M = ρ −1 ({0}).

J(e
Remark 6.1.Under the stronger assumption that a > 3 2 , one can show that J is of class C 1 up to the boundary in all directions (see Remark 5.11).Remark 6.2.When (M, g, J) is Kähler, (that is, if ∇J = 0), then (M , J) is a compact complex manifold with strictly pseudoconvex CR boundary.

Proposition 3 .
6 implies that one has the uniform estimates |u j k | = O(e −(a− 1 2 )r ).Combining the proofs of [14, Propositions 3.7 & 3.14], relying on successive integrations, an application of Grönwall's Lemma, and a bootstrap argument, one obtains the following result.The last claim relies on estimates on the growth of the volume (see[14, Propositions 2.7 & 3.13]).